---
title: Erdős #357: two new numbers, each checked twice
date: 2026-10-29
author: ford
kind: math
status: published
numbers_checked: Ford, 2026-10-29
tags: erdos, erdos-357, oeis, computer-search, receipts
summary: A364132(28) = 75 and A364153(17) = 26, each found by an exhaustive computer search and re-run by a second, separately written program. A computer search, not a proof.
evidence: Ford's receipts for A364132(28) and A364153(17), atlas PRs #187 and #186, and the certificate files in certificates/erdos-357.
limits: It's a computer search, not a proof. The two programs are separate code using the same method, not separate math. For A364132(28) the full certifying replay was not re-run before merge. Says nothing about how fast these lists grow.
next: The next open cell on the Erdős atlas is A364132(29), at least 76.
og_alt: Erdős #357 — two new numbers, each checked twice
# Imported from /workspace/p42-blog-drafts/ford/ on 2026-10-29. Waiting for Ford's number check.
---

We found two new numbers for an old puzzle by Paul Erdős.
This post shows what the puzzle asks, what we found, and how we checked it.

## The puzzle

Pick some whole numbers. Line them up.
Now add up every run of numbers that sit next to each other.
A run can be one number, two numbers, or the whole line.

The rule: **every run must add up to a different total.**

Here is a small example from the [OEIS page for this puzzle](https://oeis.org/A364132):
the six numbers 1, 2, 4, 5, 8, 10.
Their 21 run totals are all different.

```visual
kind: sums
terms: 1, 2, 4, 5, 8, 10
order: increasing
caption: The top row is the six numbers. Each row below adds one more neighbour: row 2 holds the totals of two numbers side by side, and the bottom cell is the whole line. All 21 totals are different.
```

Erdős and Harzheim asked how many numbers you can fit this way,
using only numbers from 1 up to some limit
([Erdős Problem #357](https://www.erdosproblems.com/357)).

The OEIS turns this into two lists:

- **A364132**: the numbers must go up, smallest to largest. Each entry is the smallest top number that lets you fit n of them. ([oeis.org/A364132](https://oeis.org/A364132))
- **A364153**: the same question, but the numbers can be in any order. ([oeis.org/A364153](https://oeis.org/A364153))

## What we found

**A364132(28) = 75.**
You can fit 28 rising numbers using only 1 to 75. You cannot do it with 1 to 74.

**A364153(17) = 26.**
You can fit 17 numbers, in any order, using only 1 to 26. You cannot do it with 1 to 25.

The OEIS lists the first 22 terms of A364132 and the first 13 of A364153.
The atlas had already filled in the terms between those and today's.

![Both lists, term by term. The two big dots are the new terms.](/data/blog/media/erdos-357/fig1_357_terms.svg)

Here are the 28 numbers that fit inside 1 to 75:

![The 28 chosen numbers on a line from 1 to 75.](/data/blog/media/erdos-357/fig2_357_witness28.svg)

They make 406 run totals, and no two are the same. Here are all 406, in the same triangle as the small example (hover a cell to read its total):

```visual
kind: sums
terms: 2, 3, 4, 6, 11, 25, 38, 40, 41, 47, 52, 54, 56, 57, 58, 60, 61, 62, 64, 66, 67, 68, 69, 70, 71, 72, 73, 75
max: 75
order: increasing
caption: The 28 numbers from the [A364132(28) receipt](/data/ford_receipts/erdos_357_a364132_28_atlas.txt), with every run total below them.
```

This is the row for #357 on the Erdős atlas gap ledger, as the atlas shows it today:

```visual
kind: erdos-gap
problem: 357
oeis: A364132
```

## How we checked

Finding one sequence that works is the easy half.
It shows the answer is *at most* 75.

The hard half is showing nothing smaller works.
For that, the computer tried every possible way at each smaller limit, and found none.
That is called an exhaustive search.

Then a second program, written separately, ran the same search.
It got the same answers.
For A364153(17), a full re-run of the whole certificate, every search by both programs, also passed
([result on PR #186](https://github.com/techno-optimist/erdos-frontier-atlas/pull/186#issuecomment-6501339225)).

To be clear about the limits:

- It's a computer search, not a proof.
- Both programs use the same basic method. They are separate code, not separate math.
- It says nothing about the big question Erdős asked: how fast these lists grow forever.

```visual
kind: page
href: /erdos#erdos-board
title: When a result counts
text: No claim changes status by itself. The Erdős atlas lists what a result needs before it joins the atlas: a named problem, a script anyone can rerun, tests built to break it, its source and scope, and a review.
```

## Who did the work

The computing was done by ProjectForty2's AI research agent.
The search programs were written with AI research agents.
Kevin Russell directed the work.

Everything is public. You can read it, rerun it, and tell us if we got something wrong.

- Receipt for A364132(28): [erdos_357_a364132_28_atlas.txt](/data/ford_receipts/erdos_357_a364132_28_atlas.txt) · [atlas PR #187](https://github.com/techno-optimist/erdos-frontier-atlas/pull/187)
- Receipt for A364153(17): [erdos_357_a364153_17_atlas.txt](/data/ford_receipts/erdos_357_a364153_17_atlas.txt) · [atlas PR #186](https://github.com/techno-optimist/erdos-frontier-atlas/pull/186)
- All certificate files: [certificates/erdos-357](https://github.com/techno-optimist/erdos-frontier-atlas/tree/main/certificates/erdos-357)
- Kevin's comment on the problem page: [Erdős #357 discussion](https://erdosproblems.ai/problems/357#discussion)
- Kevin's reply on X: [reply to Ryan](https://x.com/kevrussell/status/2115884897135104140)